Matjaž Krnc (2011) Fullerene graphs. EngD thesis.
Fullerene graphs are 3-connected 3-regular planar graphs with only pentagonal and hexagonal faces. The name comes from chemistry, where they can be used as a tool for modelling and analysis of carbon molecules. In this thesis, we will first make a basic overview of graph theory, where we focus on the fullerene family of graphs. Then we will analyze the structure of cyclic edge-cuts of fullerene graphs of sizes 6 and 7, distinguishing between degenerate and non-degenerate cyclic edge-cuts, regarding the arrangement of the 12 pentagons. We will prove that if there exists a non-degenerate cyclic 7-edge-cut in a fullerene graph, then the graph is a nanotube unless it is one of the two exceptions presented. We will determine that there are 57 configurations of degenerate cyclic 7-edge-cuts, and we will list all of them. In the 3rd chapter of the thesis, we will show that the diameter of a fullerene graph $G$ of order $n$ is at least Ω(√n) and at most n/5 + 1. Moreover, if G is not a (5,0)-nanotube its diameter is at most n/6 + 5/2. As a consequence, we improve the upper bound on the saturation number of fullerene graphs. We also report improved lower and upper bounds on the independence number and on the smallest eigenvalue of fullerene graphs.
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